Core reading Chapter 3 Q3.5

Hi,

Could someone please explain why the PDF is multiplied by x inside the integral when trying to verify the formula in the tables for the mean of the two parameter Pareto distribution.

Also, it would be really helpful please if you could break down the differentiation of the the CDF to get the PDF of the two parameter Pareto.

Many thanks,
Tom

 

We are deriving the mean of a continuous distribution here. The formula for which is integral of x into f(x). That's why pdf is multiplied by x. If you don't multiply it with x, you will get the sum of probability that is 1!

 

F(x)= 1- (L/L+x)^a
F(x)= 1- L^a(L+x)^-a
Pdf= F'(x) = 0 - L^a[ -a (L+x)^(-a-1)]
f(x) = (L^a × a)/ (L+x)^(a+1)

Unable to upload the pic, see if you can understand this (Used L for lembda and a for alpha).

 

That's great, thanks Bharti